Negative eigenvalue estimates for polyharmonic Schr\"odinger operators with measure-potentials: the subcritical case

Abstract

We study spectral estimates for polyharmonic Schr\"odinger operators $-\Delta^l-\mu$ in the subcritical regime $2l<\mathbf{N}$. The measure potential $\mu$ is assumed to satisfy a capacitary smallness condition which guarantees that the corresponding operator is semibounded and self-adjoint. With such a measure $\mu$ we associate an Otelbaev function, which reflects both the local concentration and the spatial distribution of the potential. In terms of this function, we obtain two-sided estimates for the distribution function of the negative eigenvalues, and derive a sufficient condition and a necessary condition for the discreteness of the negative spectrum. As an application, we establish two-sided estimates of Lieb-Thirring-type, improving the classical ones.